'They can laugh at jokes; they read every book by P.G. Wodehouse and love his quiet humour; they can enjoy a joke against you. What they cannot abide is a joke against themselves.'

I'd like to know if there is a specific word(noun, adjective) for such people. I don't know how else to further explain my question. Pretty much everything is stated in the excerpt.

Post Answer EDIT: I'm not looking for words related to sensitive-brittle natured or synonyms.
Such people enjoy joke on others but not on themselves. Maybe I'm being too specific, but such people are not at all sensitive, thin skinned or else they wouldn't be making jokes on others.

...which often carries more the sense of if you can't cope, you should leave the work to someone who can, but I wouldn't think it unusual if I heard that said to someone who complained about being the butt of too many "office banter" jokes.

I certainly can't think of one, and I think it's highly unlikely there's a single-word adjective to describe exactly the kind of person OP describes. It's just too specific.

This implies that the person is "inflexible" and prone to "snapping" at people, and probably would not enjoy being made fun of.

Humorless

Someone who doesn't like jokes much at all.

Can't take a joke

A phrase used to describe someone who does not enjoy jokes. This phrase usually implies a "practical" element to the joke, i.e. a prank.

Sensitive

Someone who reacts very strongly to mild stimuli. This would be used to describe someone who reacted with anger to you making jokes about them.

Keep in mind that almost all of these are pejoratives, and likely to make an upset person more upset if you use them. I'm sure we all have experiences of people who are very rude, and try to excuse it as "humor" and complain about your reaction as if you are "overly sensitive".

Gelotophobia is a fear of being laughed at, a type of social phobia. While most people do not like being laughed at,1 there is a sub-group of people that exceedingly fear being laughed at. Without obvious reasons, they relate laughter they hear to be directed at themselves.